# 3 different formulas of Hooke’s Law

Often we find more than one expression or formula of Hooke’s law. Here we will find out those at the same place and discuss their uses and relevance. We have a very useful post on different types of stress, strain, and elasticity modulus, that you can go through along with this post.

## Hooke’s Law For linear springs

Here we will consider a simple helical spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is Fs.

Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore.

Let x be the amount by which the free end of the spring was displaced from its initial.

Hooke’s law states that: Fs x

=> Fs = kx ……………………….. [ Hooke’s Law Equation ] (1)

where k is a positive real number, characteristic of the spring.

Moreover, the same formula holds when the spring is compressed, with Fs and x both negative in that case.

According to this formula, the graph of the applied force Fs as a function of the displacement x will be a straight line passing through the origin, whose slope is k.

## Hooke’s law equation for restoring force

Hooke’s law for spring is sometimes stated under the convention that Fs is the restoring force exerted by the spring on whatever is pulling its free end. In that case, the equation gets modified as the following equation since the direction of the restoring force is opposite to that of the displacement.

Fs = – kx …………………………… [ Hooke’s Law Equation ] (2)

## Hooke’s law for continuous media

The stresses and strains of the material inside a continuous elastic material (such as a block of rubber, the wall of a boiler, or a steel bar) are connected by a linear relationship that is mathematically similar to Hooke’s spring law and is often referred to by that name.

Hooke’s Law of Elasticity states that: For small deformations, stress is directly proportional to strain.

Stress ∝ strain

So, Stress = K. Strain   ………………….  [ Hooke’s Law Equation ] (3)
Here K is a constant.

## Summary

Here, we have seen 3 different forms of Hooke’s law equation. While solving physics numerical from this Elasticity chapter, use the right formula as per the numerical data and query.

3 different formulas of Hooke’s Law
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